Absolute continuity of real functions

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In mathematics, a real-valued function f of a real variable is absolutely continuous on a specified finite or infinite interval if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint sub-intervals y_k, k = 1, ..., n satisfies

then

Every absolutely continuous function is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous.

The Cantor function is continuous everywhere but not absolutely continuous; as is the function

on a finite interval containing the origin, or the function $f(x)=x^2$ on an infinite interval.

• If f is absolutely continuous on a finite interval then it is of bounded variation on [a,b.
• If f is absolutely continuous on the interval then it has the Luzin N property (that is, for any $L\backslash subseteq\; [a,b$ that $\backslash lambda(L)=0$, it holds that $\backslash lambda(f(L))=0$, where $\backslash lambda$ stands for the Lebesgue measure).
• If f is absolutely continuous, then f has a derivative almost everywhere.
• If f is continuous, is of bounded variation and has the Luzin N property, then it is absolutely continuous.

Absolute continuity of measures

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If μ and ν are measures on the same measure space (or, more precisely, on the same sigma-algebra) then μ is absolutely continuous with respect to ν if μ(A) = 0 for every set A for which ν(A) = 0. It is written as "μ << ν".

If μ is a signed or complex measure, it is said that μ is absolutely continuous with respect to ν if its variation |μ| satisfies |μ| << ν; equivalently, if every set A for which ν(A) = 0 is μ-null.

The Radon-Nikodym theorem states that if μ is absolutely continuous with respect to ν, and ν is σ-finite, then μ has a density, or "Radon-Nikodym derivative", with respect to ν, which implies that there exists a ν-measurable function f taking values in ^*, denoted by f = dμ/dν, such that for any ν-measurable set A we have

$\backslash mu(A)=\backslash int\_A\; f\backslash ,d\backslash nu.$

The connection between absolute continuity of real functions and absolute continuity of measures

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A measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure if and only if the point function

$F(x)=\backslash mu((-\backslash infty,x])$

is locally an absolutely continuous real function. In other words, a function is locally absolutely continuous if and only if its distributional derivative is a measure that is absolutely continuous with respect to the Lebesgue measure.

See also

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Singular measure

Reference

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Real analysis | Measure theory

Absolute Stetigkeit | Continuità assoluta | Absolute continuïteit